Nonequidistant Mesh Research Articles

Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 105 different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.

Optimal sampling design for global approximation of jump diffusion stochastic differential equations

ABSTRACTThe paper deals with strong global approximation of stochastic differential equations (SDEs) driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition (see Chapter 6.3 in [21]). We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient, with respect to asymptotic constants, than those based on the equidistant mesh.

Optimal global approximation of stochastic differential equations with additive Poisson noise

We consider strong global approximation of SDEs driven by a homogeneous Poisson process with intensity ź > 0. We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson process. We consider two classes of methods using equidistant or nonequidistant sampling of the Poisson process, respectively. We provide a construction of optimal schemes, based on the classical Euler scheme, which asymptotically attain the established minimal errors. It turns out that methods based on nonequidistant mesh are more efficient than those based on the equidistant mesh.

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

We are considering a semilinear singular perturbation reaction-diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is \(\epsilon\)-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

Analysis of the harmonic components of signal based on the values measured at nonequidistant mesh points of an equidistant mesh with a finite length that is equal to the number of residues modulo a prime

A method for nonequidistant measurements of signals on an equidistant mesh with a finite length equal to the number of residues modulo a prime p1 is proposed to determine parameters of the harmonic components of signal. The measurements are performed at the mesh points with the numbers that correspond to the primitive roots modulo a prime p1, which provides a decrease in the number of measurements and an increase in the radar capacity by a factor of 2–5. At a given number of nonequidistant measurements on the equidistant mesh with a relatively large number of mesh points, an interval of the unambiguous measurements of the longitudinal size (based on the multifrequency signal) or Doppler frequency (based on a train of single-frequency pulses) can be increased. It is demonstrated that the spectrum of signal constructed with the aid of such measurements reproduces spectral lines of harmonic oscillations contained in the signal for both coherent and incoherent signal processing. A procedure for the reconstruction of the signal at the missing equidistant mesh points using the interpolation Legendre polynomials based on the nonequidistant measurements is proposed. A further weighted processing of the reconstructed signal and the formation of spectrum using the Fourier transform will provide a decrease in the level of side lobes.

Finite Element Resistivity Forward Modeling Using Algebraic Multigrid Preconditioned Conjugate Gradient Method

An algebraic multigrid by smoothed aggregation preconditioned conjugate gradient method is developed to solve the liner system arising from 3-D direct current finite element resistivity forward modeling. The algorithm combines the efficiency of algebraic multigrid method and the stability of conjugate gradient method. Algebraic multigrid by smoothed aggregation keep in high-efficiency while simulation using local quasi-uniform mesh and its convergence effect will reduce while numerical modeling using anisotropic stretched grids. However tensor product non-equidistant mesh, a kind of anisotropic stretched grids, is often used in 3-D direct current resistivity forward modeling. In order to improve this situation, a factor is added to guide correct aggregation. Consequently, a typical example is used to prove that the improvement is the right. Finally, it is natural to conclude that the algorithm suggested in this paper is efficient and robust whether simulation using local quasi-uniform mesh or tensor product non-equidistant mesh

Variable-stepsize doubly quasi-consistent parallel explicit peer methods with global error control

Variable-stepsize explicit peer methods were designed and studied by Weiner et al. in 2008, 2009. Those schemes have proved their high efficiency in practical computations and are considered to be competitive to the best explicit Runge–Kutta embedded pairs. This paper adds more functionality to the mentioned numerical technique in terms of global error estimation and control. Theoretically, it is based on the new concept of double quasi-consistency introduced by Kulikov in 2009. This property means that the principal terms of the local and global errors coincide. In other words, the global error estimation and control can be done effectively via the conventional local error control facility, that is a standard feature of ODE solvers.Recently, Kulikov and Weiner implemented the idea of double quasi-consistency in fixed-stepsize doubly quasi-consistent parallel explicit peer methods. They also extended that result to non-equidistant meshes by an accurate polynomial interpolation technique. Here, we prove at first that the class of variable-stepsize doubly quasi-consistent methods is not empty and provide the first sample of such numerical schemes. Then, we utilize the notion of embedded formulas to evaluate and control efficiently the local error of the constructed doubly quasi-consistent peer method and, hence, its global error at the same time. Numerical examples of this paper confirm that the usual local error control implemented in doubly quasi-consistent numerical integration techniques is capable of producing numerical solutions for user-supplied accuracy conditions in automatic mode. A comparison with the third order Matlab solver ode23 is also presented.

Non-Equidistant Numerical Methods for Ordinary Differential Equation with Periodical Boundary Value

Ordinary differential equation with periodical boundary value and small parameter multiplied in the highest derivative was considered. The solution of the problem has boundary layers, which is thin region in the neighborhood of the boundary of the domain. Firstly, the properties of boundary layer were discussed. The solution was decomposed into the smooth component and the singular component. The derivatives of the smooth component and the singular component were estimated. Secondly, mesh partition techniques were presented according to one transition point method and multi-transition points method. Thirdly numerical methods based on non-equidistant mesh partition were presented to solve the problem. Finally error estimations were given for both computational methods.

Asymptotic Analysis for Hyperbolic Equation with Neumann Condition

The second-order hyperbolic equation with small parameter and Neumann con- dition is considered. This kind of problem loses the boundary conditions both in x = 0 and x = 1, while it also loses two initial boundary conditions in t = 0. The solution changes rapidly near two boundary layers and one initial layer. Firstly, the asymptotic solution was studied. The analytical solution was approximated by the degenerate solution and two boundary layer functions and one initial layer function. Secondly, three transition points were presented ac- cording to Shishkin’s idea. Non-equidistant mesh partitions both in x direction and t direction were introduced. An effective computational method is given according to non-equidistant mesh partitions. Finally, numerical experiment was given.

Numerical solution of the singularly perturbed problem with nonlocal boundary condition

Singularly perturbed boundary value problem with nonlocal conditions is examined. The appopriate solution exhibits boundary layer behavior for small positive values of the perturbative parameter. An exponentially fitted finite difference scheme on a non-equidistant mesh is constructed for solving this problem. The uniform convergence analysis in small parameter is given. Numerical example is provided, too.

Weights of the exponential fitting multistep algorithms for first-order ODEs

We describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first-order differential equations. The main novelties are that (i) we admit nonequidistant mesh points in the partition and (ii) the weights are determined on the basis of the exponential functions exp(λ ix), i=1,2,3,… rather than on the power function set, as it is done for the classical weights. In this way the method allows computing not only the weights of the well-established algorithms but also those of new ones. Another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms. Some relevant numerical illustrations are given.

Stability of Approximation Methods on Locally Non-Equidistant Meshes for Singular Integral Equations

A number of numerical methods for singular integral operators with continuous coefficients on curves with corners are studied. Necessary and sufficient conditions for stability of the methods under consideration are given. These conditions are formulated in terms of invertibility of some model operators. It is shown that the model operators belong to a well-known operator algebra, and their Fredholm properties are investigated. It is proved that the indices of the operators mentioned are equal to zero. Let Γ be a simple closed counter-clockwise oriented curve in the complex plane C. Up to now a variety of approximation methods have been proposed and investigated(see, [1], [2], [4], [7], [12]–[23]) for solving singular integral equations of Cauchy type (Ax)(τ) := a(τ)x(τ)+ b(τ) πi ∫

Fourth-order accurate difference method for the singular perturbation problem

In this paper, the numerical solution of fourth-order ordinary differential equations is considered. To approximate the differential equation, the Hermitian scheme on a special nonequidistant mesh is used. The fourth-order convergence uniform in the perturbation parameter is proved. The numerical result shows the pointwise convergence, too.

Highly Continuous Interpolants for One-Step ODE Solvers and their Application to Runge--Kutta Methods

We suggest a general method for the construction of highly continuous interpolants for one-step methods applied to the numerical solution of initial value problems of ODEs of arbitrary order. For the construction of these interpolants one uses, along with the numerical data of the discrete solution of a problem provided by a typical one-step method at endstep points, high-order derivative approximations of this solution. This approach has two main advantages. It allows an easy way of construction of high-order Runge--Kutta and Nystrom interpolants with reduced cost in additional function evaluations that also preserve the one-step nature of the underlying discrete ODE solver. Moreover, for problems which are known to possess a solution of high smoothness, the approximating interpolant resembles this characteristic, a property that on occasion might be desirable. An analysis of the stability behavior of such interpolatory processes is carried out in the general case. A new numerical technique concerning the accurate determinationof the stability behavior of numerical schemes involving higher order derivatives and/or approximations of the solution from previous grid-points over nonequidistant meshes is presented. This technique actually turns out to be of a wider interest, as it allows us to infer, in certain cases, more accurate results concerning the stability of, for example, the BDF formulas over variable stepsize grids. Moreover it may be used as a framework for analyzing more complex (and supposedly more promising) types of methods, as they are the general linear methods for first- and second-order differential equations. Many particular variants of the new method for first-order differential equations that have good prospects of finding a practical implementation are fully analyzed with respect to their stability characteristics. A detailed application concerning the construction of $C^2$ and $C^3$ continuous extensions for some fifth- and sixth-order Runge--Kutta pairs, supplemented by a detailed study of the local truncation error characteristics of a class of interpolants of this type, is also provided. Various numerical examples show, in these cases, several advantages of the newly proposed technique with respect to function evaluation cost and global error behavior, in comparison with others currently in use.

A uniform numerical method for quasilinear singular perturbation problem with a turning point

A nonlinear difference scheme is given for solving a quasilinear singularly perturbed two-point boundary value problem with a turning point. The method uses non-equidistant discretization meshes. The solution of the scheme is shown to be first order accurate in the discreteL∞ norm, uniformly in the perturbation parameter.

Defect Correction for Two-Point Boundary Value Problems on Nonequidistant Meshes

New finite difference formulae of arbitrary order are derived for the special class of second-order two-point boundary value problems $y” = f(x,y(x)), a \leq x \leq b$ . Variable mesh spacing is possible, and the required accuracy is achieved under a very mild mesh condition. A natural defect correction framework is set up to compute the higher-order approximations.

Defect correction for two-point boundary value problems on nonequidistant meshes

New finite difference formulae of arbitrary order are derived for the special class of second-order two-point boundary value problems y = f ( x , y ( x ) ) , a ≤ x ≤ b y = f(x,y(x)), a \leq x \leq b . Variable mesh spacing is possible, and the required accuracy is achieved under a very mild mesh condition. A natural defect correction framework is set up to compute the higher-order approximations.

Stormer-numerov approximation for numerical solutions of ordinary and partial differential equations

Stormer-Numerov approximations of high accuracy were developed for solutions of non-linear boundary value problems and nonlinear elliptic partial differential equations. The approximations can be easily adopted also for parabolic partial differential equations in one and more space dimensions and feature fourth-order accuracy. For boundary value problems only three nodes are necessary to obtain the desired fourth order accuracy. The finite difference formula for parabolic partial differential equations can be readily generalized to a nonequidistant mesh so that automatic regridding in space may be used. The Stomer-Numerov approximations are important for solution of problems where storage limitations and computer time expenditure preclude standard second order methods. Because of the fourth order approximations a low number of mesh points can be used for a majority of chemical engineering problems. The application of Stormer-Numerov approximations is illustrated on a number of examples.

Higher order schemes and Richardson extrapolation for singular perturbation problems

Semilinear singular perturbation problems are solved numerically by using finite–difference schemes on non-equidistant meshes which are dense in the layers. The fourth order uniform accuracy of the Hermitian approximation is improved by the Richardson extrapolation.

A compact finite difference scheme on a non-equidistant mesh

Abstract A fourth-order compact finite difference scheme is presented for second-order partial differential equations. The scheme is derived for a one-dimensional non-equidistant mesh, which makes it particularly useful in problems with sharp boundary layers. The inclusion of general boundary conditions does not reduce the order of the scheme for the boundary points. The scheme is tested on a representative model equation; we shortly discuss its stability in the simplest time-dependent heat flow problem and its use in more dimensions.